If \(f\)  is \(C^{1+\beta }\)  then we use stronger version of center bunching condition which can imply stronger results such as ergodicity and Hölder regularity of distributions \(E_f^u\oplus E_f^c\) , \(E_f^s\oplus E_f^c\)  and \(E_f^c\)  (see [1]} or [2]} for more details). A \(C^{1+\beta }\)  diffeomorphism \(f\)  is strong center bunched if there is \(\theta \in (0, \delta )\)  and functions \(\kappa , \hat{\kappa }\)  such that
\( \max \lbrace \nu ^\theta , \hat{\nu }^{\theta }\rbrace <\gamma \hat{\gamma }\) ,
\(\nu \gamma ^{-1}<\kappa ^{\theta }, \, \, \, \, \hat{\nu }\hat{\gamma }^{-1}<\hat{\kappa }^\theta ,\)  and for all \(x\in M\) , for all \(v^s\in E^s_f(x)\)  and \(v^u\in E^u_f(x)\) :
\(\kappa (x)<\Vert Df(v)\Vert , \, \, \, \,\,\, \Vert Df(v)\Vert <\hat{\kappa }(x)^{-1}.\) 
